Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

On variational methods in finite and incremental elastic deformation problems with discontinuous fields


Author: S. Nemat-Nasser
Journal: Quart. Appl. Math. 30 (1972), 143-156
DOI: https://doi.org/10.1090/qam/99733
MathSciNet review: QAM99733
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: With a view toward a numerical solution by means of the finite-element method, we give here a variational statement for large elastic deformations at finite strains which involves independent variation of the displacement, the (nonsymmetric first Piola-Kirchhoff) stress, and the deformation-gradient fields, and which includes both the boundary and the jump conditions. Then we present, for small deformations superimposed on the large, three variational statements, each involving three independent fields and each including both the boundary and the jump conditions. These statements are such that the first variation of the corresponding functional yields the field equations which characterize the equilibrium of the finitely-deformed state considered and also the field equations that pertain to the incremental deformations. Several specializations of these results are discussed. By way of illustration, finally, we present a finite-element formulation of the large deformation problem, using three independent fields, where each field is approximated by a piecewise-linear function within each element.


References [Enhancements On Off] (What's this?)

  • [1] E. Hellinger, Die allgemeinen Ansätze der Mechanik der Kontinua, Enz. math. Wis. 4, 602-694 (1914)
  • [2] C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Band III/1, Springer, Berlin, 1960 MR 0118005
  • [3] E. Reissner, On a variational theorem for finite elastic deformations, J. Math. Physics 32, 129-135 (1953) MR 0058411
  • [4] E. Koppe, Die Ableitung der Minimalprinzipien der nichtlinearen Elastizitätstheorie mittels kanonischer Transformation, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. IIa 1956, 259-266 MR 0081647
  • [5] K. Washizu, Variational methods in elasticity and plasticity, Pergamon Press, New York, 1968 MR 0391679
  • [6] E. Reissner, Variational methods and boundary conditions in shell theory, in Studies in optimization, Sympos. on Optimization, Toronto, Canada, June 11-14, 1968
  • [7] G. Prange, Habilitationsschrift (unpublished), Hannover, 1916
  • [8] E. Reissner, On a variational theorem in elasticity, J. Math. Physics 29, 90-95 (1950) MR 0037702
  • [9] Hai-Chang Hu, On some variational principles in the theory of elasticity and the theory of plasticity, Sci. Sinica 4, 33-54 (1955). (first published in Chinese in 1954)
  • [10] K. Washizu, On the variational principles of elasticity and plasticity, Technical Report #25-18, Cont. N5ori-07833, M.I.T., Cambridge, Mass. 1955
  • [11] P. M. Naghdi, On a variational theorem in elasticity and its application to shell theory, Trans. ASME Ser. E. J. Appl. Mech. 31, 647-653 (1964) MR 0173390
  • [12] W. Prager, Variational principles of linear elastostatics for discontinuous displacements, strains and stresses, in Recent progress in applied mechanics, The Folke-Odqvist Volume, edited by B. Broberg, J. Hult and F. Niordson, Almqvist and Wiksell, Stockholm, 1967, pp. 463-474
  • [13] E. Reissner, A note on variational principles in elasticity, Int. J. Solids Struct. 1, 93-95 (1965)
  • [14] M. Biot, Non-linear theory of elasticity and the linearized case for a body under initial stress, Philos. Mag. 27, 468-489 (1939)
  • [15] M. Biot, Mechanics of incremental deformations, Wiley, New York, 1965 MR 0185873
  • [16] L. Herrmann, Elasticity equations for incompressible and nearly incompressible materials by a variational theorem, AIAA J. 3, 1896-1901 (1965) MR 0184477
  • [17] T. H. Pian and P. Tong, The basis of finite element methods for solid continua, Int. J. Num. Meth. Engr. 1, 3-28 (1969)
  • [18] P. Tong, New displacement hybrid finite-element models for solid continua, Int. J. Num. Meth. Engr. 2, 73-83 (1970)
  • [19] P. Tong, An assumed stress hybrid finite element method for an incompressible and near-incompressible material, Int. J. Solids Struct. 5, 455-461 (1969)
  • [20] R. S. Dunham and K. S. Pister, A finite element application of the Hellinger--Reissner variational theorem, in Proc. second conference on matrix methods in structural mechanics, Wright--Patterson Air Force Base, 1968, pp. 471-487
  • [21] S. Nemat-Nasser and H. Shatoff, A consistent numerical method for the solution of nonlinear elasticity problems at finite strains, SIAM 20 (1971); Technical Report 2, ONR, Dept. AMES, UCSD, Jan. 1970 MR 0295667
  • [22] G. A. Wempner, Modeling of nonlinear structural systems with linear algebraic equations, UARI Research Report 86, The University of Alabama in Huntsville, Ala. 1970
  • [23] M. J. Sewell, On configuration-dependent loading, Arch. Rational Mech. Anal. 23, 327-351 (1968) MR 1553485
  • [24] S. Nemat-Nasser, On thermomechanics of elastic stability, Z. Angew. Math. Phys. 21, 538-552 (1970)
  • [25] R. Hill, On uniqueness and stability in the theory of finite elastic strain, J. Mech. Phys. Solids 5, 229-241 (1957) MR 0092379
  • [26] C. A. Felippa, Refined finite element analysis of linear and nonlinear two-dimensional structures, Ph.D. Thesis, University of California, Berkeley, Calif., 1966
  • [27] J. T. Oden and J. E. Key, Analysis of finite deformation of elastic solids by the finite element method, in IUTAM Sympos. on high speed computing of elastic structures, Liege, Belgium, 1970


Additional Information

DOI: https://doi.org/10.1090/qam/99733
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society